114 Proceedings of the Koyal Society of Edinburgh. [Sess.. 
Now if the Hamiltonian function is 
H = s x q x + s 2 q 2 + q 1 *TJ 1 cosp 1 + q&i U 4 cos (2p 1 + p 2 ) , 
where s 1 and s 2 are arbitrary, the adelphic-integral series to which we are 
led by the method of § 4 is 
Constant = s 1 q 1 - s 2 q 2 + g 1 tU 1 cos p x + 
+ 
2 Si s 2 
M 2 iu 4 cos ( 2 Pi+p 2 ) 
2Sj + s_2 
«2 TTTTg4 g »/ 4c08 (Pl+^2) , 2 COs(3j? 1 + p 2 ) \ 
! + S 2 1 l s 1 + s 2 3s x + s 2 / 
+ U 2 U q 2 0,4 f _ 3 COS (4 Pi +p 2 ) _ 6 cos (2p i -hp 2 ) 
2s 1 + s 2 \ 3sj + § 2 4s x + s 2 3sj + s 2 2 sj + s 2 
6 cos 
S l + s 2 S 2 
3^1 
i 
+UiUAA 
2(9s!+s 2 ) cos^q 4 cos (Zp l + 2p 2 ) j 
+ s 2 )(3si + s 2 ) Sj Sj + s 2 3s x + 2s 2 / 
5s x + S 2 COS^! 
2s i 4- S 2 (3-q + S 2 )(s x + S 2 ) Sj 
+ terms of the 6th and higher orders in Jq 1 and Jq 2 
(17) 
In our problem s^l, s 2 = —2, so 2s 1 +s 2 is a vanishing denominator. This 
denominator makes its appearance in the fourth term of the above expression, 
and occurs in every subsequent term, being squared in the coefficient of the 
fifth-order term g x 2 cos (2^ + p 2 ). We must now modify this series (17) 
so as to obtain an integral which has no vanishing denominators. 
In the first place, we apply the principle of § 9 : the lowest term which 
is affected by the vanishing denominator is the term 
y \ i cos ( 2p 1 +p 2 ) : 
+ s 2 
we therefore try to form an integral whose lowest term (discarding the 
non-essential factors (2s x — s 2 ) and U 4 ) shall be 
<h # cos (2^ +jp 2 ) . 
If then we suppose this integral to be 
Constant = <£=g' 1 g' 2 * cos (2p x +p 2 ) + <£ 4 + </> 5 + <£ 6 + • . . 
where cp r denotes the terms of degree r in y/q x and \/ q 2 , and substitute in 
the equation (<p, H) — 0, we find on equating to zero the terms of order 4 
that <p A is to be determined from the equation 
- 2^ = qJqJTJ i { 2 sin (p 1 + p 2 ) + sin (3^ +p 2 ) } 
op 1 cp 2 
The integral of this is 
<t>i = { 2 cos (p 1 + p 2 ) - cos (3p, +p 2 ) } 
to which, however, we may add terms of the type 
“?i 2 + /3?i?2+y^ 2 
• (is) 
